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Point vs Finite Plane

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A plane is infinite by definition, so I'm not sure what you mean by a finite plane. I assume that you actually mean some volume defined by the intersection of a series of half-spaces (i.e. the normal-facing side of a bunch of planes). In that case you just check to see if the point is in front of all the planes. If that's not what you mean then you may need to clarify a bit further.

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If you by finite plane mean a quad then you're looking for a point-polygon intersection test. These a often made by using barycentric coordinates. Google should give you plenty of results. If your quad is made of triangles google for point-triangle intersection.

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